Welcome to CS/MATH 320L – Applied Discrete Mathematics Spring 2015 Instructor: Marc Pomplun January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Instructor – Marc Pomplun Office: S-3-171 Lab: S-3-135 Office Hours: Tuesdays 4:00pm – 5:30pm Thursdays 7:00pm – 8:30pm Phone: 287-6443 (office) 287-6485 (lab) E-Mail: marc@cs.umb.edu Website: http://www.cs.umb.edu/~marc/cs320/ January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Visual Attention Lab Eye movement research January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The new EyeLink-2K System January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Example: Distribution of Visual Attention January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Selectivity in Complex Scenes January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Modeling of Brain Functions January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Modeling of Brain Functions unit and connection l a y e r l + 1 in the interpretive network unit and connection in the gating network unit and connection in the top-down bias network l a y e r l l a y e r l - 1 January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Computer Vision: January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Human-Computer Interfaces: January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Now back to CS 320L: Course Kit: Kenneth H. Rosen, Discrete Mathematics and its Applications 7th Edition (Available at the UMB Bookstore) On the Web: http://www.cs.umb.edu/~marc/cs320/ (contains all kinds of course information and also my slides in PDF and PPT formats, updated after each session) January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Your Evaluation 4 sets of exercises each set 5% 20% (only individual submissions allowed) midterm (75 minutes) 35% final exam (2.5 hours) 45% January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Grading For the assignments, exams and your course grade, the following scheme will be used to convert percentages into letter grades:  95%: A  90%: A-  86%: B+  82%: B  78%: B-  74%: C+  70%: C  66%: C-  62%: D+  56%: D  50%: D-  50%: F January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Academic Dishonesty You are allowed to discuss problems regarding your homework with other students in the class. However, you have to do the actual work (computing values, writing algorithms, drawing graphs, etc.) by yourself. You cannot copy anything from other sources (Wikipedia, other students’ work, etc.) The first violation will result in zero points for the entire homework or exam (and official notification). The second violation will result in failing the course. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Complaints about Grading If you think that the grading of your assignment or exam was unfair, write down your complaint (handwriting is OK), attach it to the assignment or exam, and give it to me or put it in my mailbox. I will re-grade the whole exam/assignment and return it to you in class. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Why Care about Discrete Math? Digital computers are based on discrete “atoms” (bits). Therefore, both a computer’s structure (circuits) and operations (execution of algorithms) can be described by discrete math. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Syllabus Logic and Set Theory Functions and Sequences Algorithms Applications of Number Theory Mathematical Reasoning Counting Probability Theory Relations and Equivalence Relations Graphs and Trees Boolean Algebra January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Mathematical Appetizers Useful tools for discrete mathematics: Logic Set Theory Functions Sequences January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Crucial for mathematical reasoning Used for designing electronic circuitry Logic is a system based on propositions. A proposition is a statement that is either true or false (not both). We say that the truth value of a proposition is either true (T) or false (F). Corresponds to 1 and 0 in digital circuits January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Statement/Proposition Game “Elephants are bigger than mice.” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? true January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Statement/Proposition Game “520 < 111” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Statement/Proposition Game “y > 5” Is this a statement? yes Is this a proposition? no Its truth value depends on the value of y, but this value is not specified. We call this type of statement a propositional function or open sentence. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Statement/Proposition Game “Today is January 29 and 99 < 5.” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Statement/Proposition Game “Please do not fall asleep.” Is this a statement? no It’s a request. Is this a proposition? no Only statements can be propositions. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Statement/Proposition Game “If elephants were red, they could hide in cherry trees.” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? probably false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

The Statement/Proposition Game “x < y if and only if y > x.” Is this a statement? yes Is this a proposition? yes … because its truth value does not depend on specific values of x and y. What is the truth value of the proposition? true January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Combining Propositions As we have seen in the previous examples, one or more propositions can be combined to form a single compound proposition. We formalize this by denoting propositions with letters such as p, q, r, s, and introducing several logical operators. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Logical Operators (Connectives) We will examine the following logical operators: Negation (NOT) Conjunction (AND) Disjunction (OR) Exclusive or (XOR) Implication (if – then) Biconditional (if and only if) Truth tables can be used to show how these operators can combine propositions to compound propositions. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Negation (NOT) Unary Operator, Symbol:  P P true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Conjunction (AND) Binary Operator, Symbol:  P Q PQ true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Disjunction (OR) Binary Operator, Symbol:  P Q PQ true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Exclusive Or (XOR) Binary Operator, Symbol:  P Q PQ true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Implication (if - then) Binary Operator, Symbol:  P Q PQ true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Biconditional (if and only if) Binary Operator, Symbol:  P Q PQ true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Statements and Operators Statements and operators can be combined in any way to form new statements. P Q P Q (P)(Q) true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Statements and Operations Statements and operators can be combined in any way to form new statements. P Q PQ  (PQ) (P)(Q) true false January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Equivalent Statements P Q (PQ) (P)(Q) (PQ)(P)(Q) true false The statements (PQ) and (P)(Q) are logically equivalent, because (PQ)(P)(Q) is always true. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Tautologies and Contradictions A tautology is a statement that is always true. Examples: R(R) (PQ)(P)(Q) If ST is a tautology, we write ST. If ST is a tautology, we write ST. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Tautologies and Contradictions A contradiction is a statement that is always false. Examples: R(R) ((PQ)(P)(Q)) The negation of any tautology is a contra- diction, and the negation of any contradiction is a tautology. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

Applied Discrete Mathematics Week 1: Logic and Sets Exercises We already know the following tautology: (PQ)  (P)(Q) Nice home exercise: Show that (PQ)  (P)(Q). These two tautologies are known as De Morgan’s laws. January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets